In this chapter, we present some principles of energy minimization that characterize equilibria in elastostatics; we also introduce two related concepts, namely the concept of duality (duality for the variational principles and duality between displacements and stresses), and the concept of variational formulations. Apart from their relevance to mechanics, these concepts also play an important role in the numerical approximation of the problems we consider (but we do not address this here of course), and, in other related forms, in many other areas of science.

Elastic energy of a material

In an isotropic elastic material, we can define a quadratic functional w = w(ε), called the energy function, such that:

for every tensor field ε, and thus Eq. (15.1) is then identical to the stress–strain law of the material. This function is thus defined by

hence,

or else

Consequently, w is a positive definite quadratic form in ε (we will always assume that μ and κ = λ + 2μ/3 are strictly positive).

If we now consider a tensor field ε defined on, we can associate with it the functional

We have the following result, which provides a physical interpretation of W and justifies its introduction:

Proposition 15.1.We consider an elastic system S that is in its natural state (ε = 0) and fills the volume Ω at time t = 0 and that is at time t = t1in the deformation state corresponding to the deformation tensor ε. Then, –W(ε) is, to first approximation, the work of the internal forces between times 0 and t1. Thus, W(ε) is the amount of energy stored in stresses between 0 and t1.